The recent discovery of the C60 "Buckminsterfullerene" and a host of other cage carbon molecules has ushered in a new branch of chemistry with quite mind-boggling potential. The arrangement of the carbon atoms and their relative positions at vertices in these structures trace out some interesting polyhedra.

As early as 1988 Harold Kroto, a co-discoverer of the "buckyball" stated
in Space, Stars, C

and Soot: The intriguing revelation that 12 pentagonal "defects" convert
a plane hexagonal array of any size into a quasi-icosahedral cage .....
The implication of this statement is that there are an almost infinite
number of discrete molecules of this sort possible. In fact very quickly
C28, C32, C50, C70, C76, and C84 were elucidated. I humbly suggested to
Harold Kroto that the C80 because of its unsurpassed sphericity had surely
been overlooked; he replied that it was possible and its discovery was
awaiting some bright young chemist. (Kroto was subsequently knighted and
received the Nobel Prize for his role in the Buckyball" saga). However,
in purely geometric terms the proposed C80 is an equilibrium combination
of the rhombic triacontahedron and the pentagonal dodecahedron while the
C60, in these terms, is an equilibrium combination between trigonal icosahedron
and pentagonal dodecahedron.

Fullerenes consist essentially of hexagonal carbon rings (benzene) linked
to each other partly via pentagons. The relationship between the number
of apices (a, carbon atoms) and hexagon carbon rings (n) (pentagon rings
always number 12) is given by: a = 2(n + 10). This will indicate the fullerenes
theoretically possible. Although each of these possibilities does represent
a definite polyhedron, a large number of them have relatively low symmetry.
Transposing the relationship above:

n = a/2 - 10.

Theoretically the fullerenes form molecules as the examples below. Each
successive fullerene involves the addition of one benzene ring which introduces
two additional vertices (carbon atom locations) since each vertex is shared
by three rings.

n : 0 - C20 (Not a fullerene but the pentagonal dodecahedron) | |

n : 1 - C22 | n : 2 - C24 |

n : 3 - C26 | n : 10 - C40 |

n : 15 - C50 | n : 20 - C60 |

n : 25 - C70 | n : 30 - C80 |

n : 38 - C96 | n : 50 - C120 |

n : 100 - C220 | n : 115 - C240, etc. |

**Combinations of the three hexoctahedral symetry forms : the pentagonal
dodecahedron ( top most figure in red), the trigonal icosahedron
(bottom right corner figure in yellow), and the rhombic triacontrahedron
(bottom left hand corner figure in blue). The intermediate forms
represent various combinations of the two basic structures only; the four
forms in the centre are composed of all three fundamental arrays in different
proportions.**

Only those proposed molecules with relatively low stress and distortion will be stable. Pre-eminent among these is C60 original "buckyball" whose existence was predicted before its discovery. Polyhedrally it consists of 12 regular pentagons faces and 20 regular hexagon faces - in fact an equilibrium combination of the regular pentagonal dodecahedron and the regular trigonal icosahedron which we have encountered several times before : each carbon atom (located at an apex) has precisely the same environment as the others and identical bonding and hence the great stability of the molecule. This is the smallest fullerene molecule having full hexicosahedral class symmetry and has been referred to as the most spherical molecule theoretically possible (Kroto, 1988). This is possibly not true. This author's postulated C80 consists of 12 regular pentagons and 30 hexagons (not quite regular, having two internal angles of 116.565 and four of 121.717 degrees) but is approximately 11% more spherical than C60. This postulated C80 is remarkable in that polyhedrally it is an equilibrium combination of pentagonal dodecahedron and rhombic triacontahedron and has full isometric pentagonal symmetry but does not fall in the fullerene series as outlined above. The author has identified this polyhedral combination as the form of the herpes virus and it has also been patented as an alternative soccer football design.

Those postulated fullerene molecules that, polyhedrally, have full hexicosahedral
symmetry fall in a series that includes C60 and have the general formula
60x^{2} where x is an integer. The addition of a single ring drops
the symmetry dramatically and full symmetry is again attained only by the
next member of the series. The first few members of the series are:

60.1^{2 }: C60 ; 60.2^{2} : C240 ; 60.3^{2}:
C540

60.4^{2} : C960 ; 60.5^{2}: C1500 ; 60.6^{2}:
C2160

Apparently the mix of fullerenes first discovered consisted dominantly of the C60 molecule together with a significant amount of the C70 and very much smaller quantities of a number of others. However the C80,specifically, was not detected and, to this day, April 1996, has not been found as far as can be established. This polyhedral form (equilibrium combination of the dodecahedron and icosahedron - C80) is also that of the herpes virus and has been patented as an alternate to the soccer football.

The tendency is that the larger the molecule (directly related to the
number of carbon atoms) the more closely the structure approaches a true
icosahedron with the pentagons forming the twelve vertices. This means
that the transition from low fullerenes to the giant ones represents, to
some extent, the transition from pentagonal dodecahedron to trigonal icosahedron
which is shown in the figure. C20 with no hexagons represents the dodec.
However, apparently the strain energy involved in such a carbon structure
would be such that it is unlikely to exist in practice. The giant theoretical
fullerines start approaching the icosahedron in form with the ever relative
diminution of the pentagons forming the apices and representing the dodec.
The proposed C80 fullerene, on the other hand, lies on the transition between
pentagonal dodecahedron and rhombic triacontahedron and represents the
equilibrium combination of these two forms. The hexagonal face distortion
in this structure is relatively small and the sphericity of this equilibrium
combination is higher than that of the equilibrium combination between
dodec and icosahedron (C60) hence my letter to Professor Kroto. If this
molecule is discovered it will not only displace the buckminsterfullerene
as the roundest molecule known but the series based thereon approaches
a true sphere in contrast to the generally recognized series which, as
indicated above, approaches a trigonal icosahedron. This new suggested
series also has full hexicosahedral symmetry and has the general formula
in respect of apices (carbon atoms) 20x^{2} where x is an integer.
The first few members of the series are:

20.1^{2 }: C20 ; 20.2^{2 }: C80 ; 20.3^{2 }:
C180 ;20.4^{2 }: C320

20.5^{2 }: C500 ; 20.6^{2 }: C720 ; 20.7^{2 }:
C980 ; 20.8^{2} : C1280

Retaining their symmetry, these increasingly spherical hypothetical molecules of the series appear to have no imbalances and no icosahedral "edges" around which the carbon rings need to bend unlike is the case in the "buckyball" series. Besides, the pentagon isolation principle (according to which no pentagons should abut each other) is fulfilled. A number of scientific reasons have been put forward to account for the preference of icosahedral fullerenes rather than spherical ones. The point made that the bonds near the pentgons being somewhat variable in length is borne out by this author's diagrams (Fig. 10.6). Nevertheless, perhaps the spherical series fullerenes will be discovered or synthesized one day. Each member of the series constitutes polyhedrally an identifiable combination of the seven convex forms of the hexicosahedral class. As the molecules become larger and larger plurality of the variable forms (especially the hexicosahedron itself) takes place on an increasing basis. The three fixed forms, namely, the dodecahedron, icosahedron and triacontahedron, of course, remain unique